# Prandtl–Glauert singularity

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For the condensation cloud that forms around objects moving at transonic speeds, see vapor cone.
Example of a Jet producing a shockwave at transonic speeds, creating a vapor cone. The Prandtl–Glauert singularity was incorrectly predicted to occur under these conditions. Jet F/A-18.

The Prandtl–Glauert singularity is the prediction by the Prandtl–Glauert transformation that infinite pressures would be experienced by an aircraft as it approaches the speed of sound. Because it is invalid to apply the transformation at these speeds, the predicted singularity does not emerge. This is related to the early 20th century misconception of the impenetrability of the sound barrier.

## Reasons of invalidity around Mach 1

Near the sonic speed (M=1) the transformation features a singularity, although this point is not within the area of validity. The singularity is also called the Prandtl–Glauert singularity, and the aerodynamic forces are calculated to approach infinity. In reality, the aerodynamic and thermodynamic perturbations do get amplified strongly near the sonic speed, but they remain finite and a singularity does not occur. An explanation for this is that the Prandtl–Glauert transformation is a linearized approximation of compressible, inviscid potential flow. As the flow approaches sonic speed, the nonlinear phenomena dominate within the flow, which this transformation completely ignores for the sake of simplicity.

## Prandtl-Glauert transformation

Plot of the Prandtl-Glauert transformation as a function of Mach number. Notice the infinite limit at Mach 1.

The Prandtl-Glauert transformation is found by linearizing the potential equations associated with compressible, inviscid flow. For two-dimensional flow, the linearized pressures in such a flow are equal to those found from incompressible flow theory multiplied by a correction factor. This correction factor is given below:[1]

$c_{p} = \frac {c_{p0}} {\sqrt {|1-{M_{\infty}}^2|}}$

where

This formula is known as "Prandtl's rule", and works well up to low-transonic Mach numbers (M < ~0.7). However, note the limit:

$\lim_{M_{\infty} \to 1 }c_p = \infty$

This obviously nonphysical result (of an infinite pressure) is known as the Prandtl–Glauert singularity.

## References

1. ^ Erich Truckenbrodt: Fluidmechanik Band 2, 4. Auflage, Springer Verlag, 1996, p. 178-179